I agree that I've never heard of a service to do this, probably because there's no money in it. Most computers come with a CD drive that can read and write. And blank CD's are typically less than $10 for a pack of 50 or 100.
However, there are different ways you can get music recommendations. I personally use Pandora, Spotify, and Last.fm on a regular basis
YouTube - put the name of your favorite songs and artist in and on the right side, you'll get a list of recommended tracks. Some of them are different versions, but you'll also get some new music in there as well.
Pandora - I prefer Pandora's music matching algorithm to Spotify's. You can create a station using your favorite song and then add variety with your favorites. For example, you'd start the station with the first song on your list...then go to add variety and add the additional songs on your list. As you listen to the station, you can thumb up your favorites or thumb down to get rid of the song. As you thumb up the algorithm will learn your preferences and do a better job of recommending music. I've found some really great artists I like that way.
Spotify - Does not allow you to add variety to your radio stations, so you'd have to create a separate station for each song and then move your favorites to a playlist.
Last.fm - will track your listening history across a bunch of different applications including iTunes, spotify, and I think Pandora as well. Once you've listened to enough music, it will start to recommend music you might like.
There are others like 8tracks and Gnoostic that also recommend music...but I haven't used them so can't attest to their helpfulness.